The present invention relates to a method for displaying a three-dimensional surface.
Recently, it has been a common practice to display a three-dimensional surface on a two-dimensional screen of a graphic display unit.
In a prior art method, a curve (to be referred to as a characteristic curve herebelow) representing an edge constituting a three-dimensional surface to be generated is moved by various distances along two curves (to be referred to as control curves herebelow) enclosing the characteristic curve so as to obtain a plurality of curves (intermediate curves) to be found on the surface, thereby displaying the three-dimensional surface with the intermediate curves.
For example, according to the interpolation proposed by Coons (Massachusetts Institute of Technology, Project MAC Technical Report, MAC-TR-41 (1967. 6), intermediate curves are generated only by effecting the parallel displacement and blending of the characteristic curve.
The characteristic curve is independently moved along the first control curve and a second control curve, respectively to generate two intermediate curves, which are then blended to be an intermediate curve linking the two control curves. That is, the characteristic curve is parallelly moved along the first control curve with an end of the characteristic curve kept stayed on the first control curve, which generates the first intermediate curve. Similarly, the characteristic curve is moved along the second control curve with an end thereof kept stayed on the second control curve so as to generate the second intermediate curve. As a consequence, although an end of the first intermediate curve and an end of the second intermediate curve exist on the first and second control curves, respectively; the other ends of the first and second intermediate curves are not found on the first and second control curves. A weighted mean of these two intermediate curves are calculated, or these lines are blended, by use of an appropriate weight function, thereby attaining an intermediate curve having the ends on the first and second control curves, respectively.
In this method, however, .circle.1 the intermediate curves can be generated only by parallelly moving a characteristic curve, and hence the kinds of intermediate curves to be generated are limited; furthermore, .circle.2 the first and second intermediate curves are independently generated, which causes the intermediate curves to be changed according to the blended values; and hence the intermediate curves often include undulated portions. Consequently, the surface represented by use of these intermediate curves also contains such undulated portions. Due to the drawbacks, the interpolation method proposed by Coons can only be applied to a situation where only small surface patches are generated; and a great amount of point series data is required to be inputted when generating a large surface, for example, of an outer surface of an industrial product.
To remove the first disadvantageous feature of the Coons' method, there has been known a method in which such operations as rotation, enlargement, and shrinkage are applied to the characteristic curve in addition to the parallel displacement (Proceedings of Nat. Conv. IECE Japan (1977), Vol. 5, p. 294; Proceedings of 29th Nat. Conv. of the Information Processing Society of Japan, Vol. 5, pp. 1749-1750). In this method, however, a characteristic curve is independently moved along the first and second control curves, so that an intermediate curve is generated by blending the resultant curves; however, the second drawback item identified above, namely, the undulation of the surface is not removed.
In addition, according to the Japanese patent unexamined publication No. 57-5109, the control and characteristic curves each are limited to a plane curve; consequently the kinds of the curves to be generated are restricted.